ODE
\[ -y(x) \left (n (n+1)-a^2 x^2\right )+x^2 y''(x)+2 x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0206893 (sec), leaf count = 22
\[\left \{\left \{y(x)\to c_1 j_n(a x)+c_2 y_n(a x)\right \}\right \}\]
Maple ✓
cpu = 0.062 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{{\sl Y}_{n+{\frac {1}{2}}}\left (ax\right )}+{\it \_C1}\,{{\sl J}_{n+{\frac {1}{2}}}\left (ax\right )} \right ) {\frac {1}{\sqrt {x}}}} \right \} \] Mathematica raw input
DSolve[-((n*(1 + n) - a^2*x^2)*y[x]) + 2*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*SphericalBesselJ[n, a*x] + C[2]*SphericalBesselY[n, a*x]}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-(n*(n+1)-a^2*x^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*BesselY(n+1/2,a*x)+_C1*BesselJ(n+1/2,a*x))/x^(1/2)